Spectral theory and dynamics on hyperbolic manifolds

双曲流形的谱理论和动力学

基本信息

批准号：
1401747
负责人：
Dubi Kelmer
金额：
$ 15.4万
依托单位：
Boston College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-09-01 至 2017-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401747&HistoricalAwards=false
关键词：
Spectral theory dynamics hyperbolic manifolds

项目摘要

Many problems in number theory (for instance, understanding the number and geometry of integer solutions to polynomial equations) can be approached by studying dynamics of certain flows on symmetric spaces. On the other hand, classical tools from analytic number theory can be adapted and used to better understand the geometry and dynamics of these spaces. This project investigates a number of problems where spectral theory and tools from analytic number theory are used to study geometry and dynamics of certain group actions on hyperbolic manifolds. Dynamics on hyperbolic manifolds come up in many areas of mathematics: In number theory they are central in the theory of automorphic forms; in low dimensional topology they are instrumental in the proof of the geometrization theorem; in dynamics the geodesic flow on hyperbolic manifolds is a classic example of hyperbolic (Anosov) dynamical systems; in mathematical physics the spectrum of hyperbolic manifolds can be viewed as an example of quantized chaotic systems. This project investigates a number of problems where spectral theory and tools from analytic number theory are used to study geometry and dynamics of certain group actions on hyperbolic manifolds, with and without arithmetic structure. Specifically, the research studies questions regarding the length spectrum of closed geodesics to understand how much of the geometry of a hyperbolic manifold can be obtained from information on the lengths of its closed geodesics; questions on the rate of escape to infinity of one parameter flows on hyperbolic spaces (originating from problems in Diophantine approximations); and questions on the strong spectral gap property on locally symmetric spaces, which is crucial to many applications to hyperbolic dynamics and number theory.

数量理论中的许多问题（例如，可以通过研究对称空间上某些流的动力学来解决整数解决方案的数量和几何形状）。另一方面，可以对分析数理论的经典工具进行调整，并用来更好地理解这些空间的几何形状和动力学。该项目研究了许多问题，其中使用分析数理论的光谱理论和工具用于研究某些小组作用在双曲线歧管上的几何形状和动态。双曲线歧管上的动力学在许多数学领域都出现：在数量理论中，它们在自动形式理论中是核心。在低维拓扑中，它们在几何定理的证明中发挥了作用。在动力学中，双曲线歧管上的大地测量流是双曲线（Anosov）动力学系统的经典示例。在数学物理学中，双曲歧管的光谱可以看作是量化混乱系统的一个例子。该项目调查了许多问题，其中使用或不具有算术结构，使用分析数理论中的光谱理论和工具来研究某些小组作用在双曲线歧管上的几何形状和动态。具体而言，研究研究问题有关封闭的大地测量学的长度光谱的疑问，以了解从有关其封闭的大地测量学长度的信息中获得多少双曲线歧管的几何形状；关于双曲线空间上一个参数流的逃生速率的问题（起源于Diophantine近似中的问题）；以及有关局部对称空间上强频谱差距的问题，这对于多余的动力学和数字理论至关重要。